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Deep Learning · Attention Mechanisms

Yancheng Wang, Ping Li, Yingzhen Yang

Self-attention and transformers have been widely used in deep learning. Recent efforts have been devoted to incorporating transformer blocks into different types of neural architectures, including those with convolutions, leading to various visual transformers for computer vision tasks. In this paper, we propose a novel and compact transformer block, Transformer with Differentiable Channel Selection, or DCS-Transformer. DCS-Transformer features channel selection in the computation of the attention weights and the input/output features of the MLP in the transformer block. Our DCS-Transformer is compatible with many popular and compact transformer networks, such as MobileViT and EfficientViT, and it reduces the FLOPs of the visual transformers while maintaining or even improving the prediction accuracy. In the experiments, we replace all the transformer blocks in MobileViT and EfficientViT with DCS-Transformer blocks, leading to DCS-Transformer networks with different backbones. The DCS-Transformer is motivated by reduction of Information Bottleneck, and a novel variational upper bound for the IB loss which can be optimized by SGD is derived and incorporated into the training loss of the network with DCS-Transformer. Extensive results on image classification and object detection evidence that DCS-Transformer renders compact and efficient visual transformers with comparable or much better prediction accuracy than the original visual transformers. The code of DCS-Transformer is available at https://github.com/Statistical-Deep-Learning/DCS-Transformer.

Optimization · Zero-order and Black-box Optimization

Ruigang Wang, Krishnamurthy Dvijotham, Ian Manchester

This paper presents a new *bi-Lipschitz* invertible neural network, the BiLipNet, which has the ability to smoothly control both its *Lipschitzness* (output sensitivity to input perturbations) and *inverse Lipschitzness* (input distinguishability from different outputs). The second main contribution is a new scalar-output network, the PLNet, which is a composition of a BiLipNet and a quadratic potential. We show that PLNet satisfies the Polyak-Łojasiewicz condition and can be applied to learn non-convex surrogate losses with a unique and efficiently-computable global minimum. The central technical element in these networks is a novel invertible residual layer with certified strong monotonicity and Lipschitzness, which we compose with orthogonal layers to build the BiLipNet. The certification of these properties is based on incremental quadratic constraints, resulting in much tighter bounds than can be achieved with spectral normalization. Moreover, we formulate the calculation of the inverse of a BiLipNet -- and hence the minimum of a PLNet -- as a series of three-operator splitting problems, for which fast algorithms can be applied.

Deep Learning

Behrooz Tahmasebi, Ashkan Soleymani, Dara Bahri, Stefanie Jegelka, Patrick Jaillet

Recently, there has been a surge in interest in developing optimization algorithms for overparameterized models as achieving generalization is believed to require algorithms with suitable biases. This interest centers on minimizing sharpness of the original loss function; the Sharpness-Aware Minimization (SAM) algorithm has proven effective. However, most literature only considers a few sharpness measures, such as the maximum eigenvalue or trace of the training loss Hessian, which may not yield meaningful insights for non-convex optimization scenarios like neural networks. Additionally, many sharpness measures are sensitive to parameter invariances in neural networks, magnifying significantly under rescaling parameters. Motivated by these challenges, we introduce a new class of sharpness measures in this paper, leading to new sharpness-aware objective functions. We prove that these measures are *universally expressive*, allowing any function of the training loss Hessian matrix to be represented by appropriate hyperparameters. Furthermore, we show that the proposed objective functions explicitly bias towards minimizing their corresponding sharpness measures, and how they allow meaningful applications to models with parameter invariances (such as scale-invariances). Finally, as instances of our proposed general framework, we present *Frob-SAM* and *Det-SAM*, which are specifically designed to minimize the Frobenius norm and the determinant of the Hessian of the training loss, respectively. We also demonstrate the advantages of our general framework through extensive experiments.

Li Sun, Zhenhao Huang, Hao Peng, YuJie Wang, Chunyang Liu, Philip Yu

Graph clustering is a fundamental problem in machine learning. Deep learning methods achieve the state-of-the-art results in recent years, but they still cannot work without predefined cluster numbers. Such limitation motivates us to pose a more challenging problem of graph clustering with unknown cluster number. We propose to address this problem from a fresh perspective of graph information theory (i.e., structural information). In the literature, structural information has not yet been introduced to deep clustering, and its classic definition falls short of discrete formulation and modeling node features. In this work, we first formulate a differentiable structural information (DSI) in the continuous realm, accompanied by several theoretical results. By minimizing DSI, we construct the optimal partitioning tree where densely connected nodes in the graph tend to have the same assignment, revealing the cluster struc- ture. DSI is also theoretically presented as a new graph clustering objective, not requiring the pre-defined cluster number. Furthermore, we design a neural LSEnet in the Lorentz model of hyperbolic space, where we integrate node features to structural information via manifold-valued graph convolution. Extensive empirical results on real graphs show the superiority of our approach.

Optimization · Non-Convex

Zakhar Shumaylov, Jeremy Budd, Subhadip Mukherjee, Carola-Bibiane Schönlieb

Variational regularisation is the primary method for solving inverse problems, and recently there has been considerable work leveraging deeply learned regularisation for enhanced performance. However, few results exist addressing the convergence of such regularisation, particularly within the context of critical points as opposed to global minimisers. In this paper, we present a generalised formulation of convergent regularisation in terms of critical points, and show that this is achieved by a class of weakly convex regularisers. We prove convergence of the primal-dual hybrid gradient method for the associated variational problem, and, given a Kurdyka-Łojasiewicz condition, an $\mathcal{O}(\log{k}/k)$ ergodic convergence rate. Finally, applying this theory to learned regularisation, we prove universal approximation for input weakly convex neural networks (IWCNN), and show empirically that IWCNNs can lead to improved performance of learned adversarial regularisers for computed tomography (CT) reconstruction.

Deep Learning · Theory

Guohao Shen

We investigate the complexity of deep neural networks through the lens of functional equivalence, which posits that different parameterizations can yield the same network function. Leveraging the equivalence property, we present a novel bound on the covering number for deep neural networks, which reveals that the complexity of neural networks can be reduced. Additionally, we demonstrate that functional equivalence benefits optimization, as overparameterized networks tend to be easier to train since increasing network width leads to a diminishing volume of the effective parameter space. These findings can offer valuable insights into the phenomenon of overparameterization and have implications for understanding generalization and optimization in deep learning.

Deep Learning

Zi-Hao Qiu, Siqi Guo, Mao Xu, Tuo Zhao, Lijun Zhang, Tianbao Yang

The temperature parameter plays a profound role during training and/or inference with large foundation models (LFMs) such as large language models (LLMs) and CLIP models. Particularly, it adjusts the logits in the softmax function in LLMs, which is crucial for next token generation, and it scales the similarities in the contrastive loss for training CLIP models. A significant question remains: `` Is it viable to learn a neural network to predict a personalized temperature of any input data for enhancing LFMs?" In this paper, we present a principled framework for learning a small yet generalizable temperature prediction network (TempNet) to improve LFMs. Our solution is composed of a novel learning framework with robust losses underpinned by constrained distributionally robust optimization (DRO), and a properly designed TempNet with theoretical inspiration. TempNet can be trained together with a large foundation model from scratch or learned separately given a pretrained foundation model. It is not only useful for predicting personalized temperature to promote the training of LFMs but also generalizable and transferable to new tasks. Our experiments on LLMs and CLIP models demonstrate that TempNet greatly improves the performance of existing solutions or models.

Deep Learning

Amit Peleg, Matthias Hein

Neural networks typically generalize well when fitting the data perfectly, even though they are heavily overparameterized. Many factors have been pointed out as the reason for this phenomenon, including an implicit bias of stochastic gradient descent (SGD) and a possible simplicity bias arising from the neural network architecture. The goal of this paper is to disentangle the factors that influence generalization stemming from optimization and architectural choices by studying *random* and *SGD-optimized* networks that achieve zero training error. We experimentally show, in the low sample regime, that overparameterization in terms of increasing width is beneficial for generalization, and this benefit is due to the bias of SGD and not due to an architectural bias. In contrast, for increasing depth, overparameterization is detrimental for generalization, but random and SGD-optimized networks behave similarly, so this can be attributed to an architectural bias.

Probabilistic Methods · Bayesian Models and Methods

Luis A. Ortega, Simon Rodriguez Santana, Daniel Hernández-Lobato

The Linearized Laplace Approximation (LLA) has been recently used to perform uncertainty estimation on the predictions of pre-trained deep neural networks (DNNs). However, its widespread application is hindered by significant computational costs, particularly in scenarios with a large number of training points or DNN parameters. Consequently, additional approximations of LLA, such as Kronecker-factored or diagonal approximate GGN matrices, are utilized, potentially compromising the model's performance. To address these challenges, we propose a new method for approximating LLA using a variational sparse Gaussian Process (GP). Our method is based on the dual RKHS formulation of GPs and retains as the predictive mean the output of the original DNN. Furthermore, it allows for efficient stochastic optimization, which results in sub-linear training time in the size of the training dataset. Specifically, its training cost is independent of the number of training points. We compare our proposed method against accelerated LLA (ELLA), which relies on the Nyström approximation, as well as other LLA variants employing the sample-then-optimize principle. Experimental results, both on regression and classification datasets, show that our method outperforms these already existing efficient variants of LLA, both in terms of the quality of the predictive distribution and in terms of total computational time.

Optimization · Non-Convex

Mimoun Mohamed, Francois Malgouyres, Valentin Emiya, Caroline Chaux

The *straight-through estimator* (STE) is commonly used to optimize quantized neural networks, yet its contexts of effective performance are still unclear despite empirical successes. To make a step forward in this comprehension, we apply STE to a well-understood problem: *sparse support recovery*. We introduce the *Support Exploration Algorithm* (SEA), a novel algorithm promoting sparsity, and we analyze its performance in support recovery (a.k.a. model selection) problems. SEA explores more supports than the state-of-the-art, leading to superior performance in experiments, especially when the columns of $A$ are strongly coherent. The theoretical analysis considers recovery guarantees when the linear measurements matrix $A$ satisfies the *Restricted Isometry Property* (RIP). The sufficient conditions of recovery are comparable but more stringent than those of the state-of-the-art in sparse support recovery. Their significance lies mainly in their applicability to an instance of the STE.

Deep Learning · Graph Neural Networks

Rui Miao, Kaixiong Zhou, Yili Wang, Ninghao Liu, Ying Wang, Xin Wang

Graph neural networks (GNNs) have exhibited prominent performance in learning graph-structured data. Considering node classification task, based on the i.i.d assumption among node labels, the traditional supervised learning simply sums up cross-entropy losses of the independent training nodes and applies the average loss to optimize GNNs' weights. But different from other data formats, the nodes are naturally connected. It is found that the independent distribution modeling of node labels restricts GNNs' capability to generalize over the entire graph and defend adversarial attacks. In this work, we propose a new framework, termed joint-cluster supervised learning, to model the joint distribution of each node with its corresponding cluster. We learn the joint distribution of node and cluster labels conditioned on their representations, and train GNNs with the obtained joint loss. In this way, the data-label reference signals extracted from the local cluster explicitly strengthen the discrimination ability on the target node. The extensive experiments demonstrate that our joint-cluster supervised learning can effectively bolster GNNs' node classification accuracy. Furthermore, being benefited from the reference signals which may be free from spiteful interference, our learning paradigm significantly protects the node classification from being affected by the adversarial attack.

Deep Learning · Attention Mechanisms

Siqi Miao, Zhiyuan Lu, Mia Liu, Javier Duarte, Pan Li

This study introduces a novel transformer model optimized for large-scale point cloud processing in scientific domains such as high-energy physics (HEP) and astrophysics. Addressing the limitations of graph neural networks and standard transformers, our model integrates local inductive bias and achieves near-linear complexity with hardware-friendly regular operations. One contribution of this work is the quantitative analysis of the error-complexity tradeoff of various sparsification techniques for building efficient transformers. Our findings highlight the superiority of using locality-sensitive hashing (LSH), especially OR & AND-construction LSH, in kernel approximation for large-scale point cloud data with local inductive bias. Based on this finding, we propose LSH-based Efficient Point Transformer (**HEPT**), which combines E$^2$LSH with OR & AND constructions and is built upon regular computations. HEPT demonstrates remarkable performance on two critical yet time-consuming HEP tasks, significantly outperforming existing GNNs and transformers in accuracy and computational speed, marking a significant advancement in geometric deep learning and large-scale scientific data processing. Our code is available at https://github.com/Graph-COM/HEPT.

Reinforcement Learning · Function Approximation

Yiming Meng, Ruikun Zhou, Amartya Mukherjee, Maxwell Fitzsimmons, Christopher Song, Jun Liu

Solving nonlinear optimal control problems is a challenging task, particularly for high-dimensional problems. We propose algorithms for model-based policy iterations to solve nonlinear optimal control problems with convergence guarantees. The main component of our approach is an iterative procedure that utilizes neural approximations to solve linear partial differential equations (PDEs), ensuring convergence. We present two variants of the algorithms. The first variant formulates the optimization problem as a linear least square problem, drawing inspiration from extreme learning machine (ELM) for solving PDEs. This variant efficiently handles low-dimensional problems with high accuracy. The second variant is based on a physics-informed neural network (PINN) for solving PDEs and has the potential to address high-dimensional problems. We demonstrate that both algorithms outperform traditional approaches, such as Galerkin methods, by a significant margin. We provide a theoretical analysis of both algorithms in terms of convergence of neural approximations towards the true optimal solutions in a general setting. Furthermore, we employ formal verification techniques to demonstrate the verifiable stability of the resulting controllers.

Deep Learning · Algorithms

Hanna Mazzawi, Xavi Gonzalvo, Michael Wunder, Sammy Jerome, Benoit Dherin

Training deep neural networks for large language models (LLMs) remains computationally very expensive. To mitigate this, network growing algorithms offer potential cost savings, but their underlying mechanisms are poorly understood. In this paper, we propose a theoretical framework using backward error analysis to illuminate the dynamics of mid-training network growth. Furthermore, we introduce Deep Fusion, an efficient network training approach that leverages pre-trained initializations of smaller networks, facilitating network growth from diverse sources. Our experiments validate the power of our theoretical framework in guiding the optimal use of Deep Fusion. With carefully optimized training dynamics, Deep Fusion demonstrates significant reductions in both training time and resource consumption. Importantly, these gains are achieved without sacrificing performance. We demonstrate reduced computational requirements, and improved generalization performance on a variety of NLP tasks and T5 model sizes.

Optimization · Discrete and Combinatorial Optimization

Jaron Maene, Vincent Derkinderen, Luc De Raedt

The limitations of purely neural learning have sparked an interest in probabilistic neurosymbolic models, which combine neural networks with probabilistic logical reasoning. As these neurosymbolic models are trained with gradient descent, we study the complexity of differentiating probabilistic reasoning. We prove that although approximating these gradients is intractable in general, it becomes tractable during training. Furthermore, we introduce *WeightME*, an unbiased gradient estimator based on model sampling. Under mild assumptions, WeightME approximates the gradient with probabilistic guarantees using a logarithmic number of calls to a SAT solver. Lastly, we evaluate the necessity of these guarantees on the gradient. Our experiments indicate that the existing biased approximations indeed struggle to optimize even when exact solving is still feasible.

Applications · Robotics

Yunhao Luo, Chen Sun, Josh Tenenbaum, Yilun Du

Effective motion planning in high dimensional spaces is a long-standing open problem in robotics. One class of traditional motion planning algorithms corresponds to potential-based motion planning. An advantage of potential based motion planning is composability -- different motion constraints can easily combined by adding corresponding potentials. However, constructing motion paths from potentials requires solving a global optimization across configuration space potential landscape, which is often prone to local minima. We propose a new approach towards learning potential based motion planning, where we train a neural network to capture and learn an easily optimizable potentials over motion planning trajectories. We illustrate the effectiveness of such approach, significantly outperforming both classical and recent learned motion planning approaches and avoiding issues with local minima. We further illustrate its inherent composability, enabling us to generalize to a multitude of different motion constraints. Project website at https://energy-based-model.github.io/potential-motion-plan.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Andrew Lowy, Jonathan Ullman, Stephen Wright

We provide a simple and flexible framework for designing differentially private algorithms to find approximate stationary points of non-convex loss functions. Our framework is based on using a private approximate risk minimizer to "warm start" another private algorithm for finding stationary points. We use this framework to obtain improved, and sometimes optimal, rates for several classes of non-convex loss functions. First, we obtain improved rates for finding stationary points of smooth non-convex empirical loss functions. Second, we specialize to quasar-convex functions, which generalize star-convex functions and arise in learning dynamical systems and training some neural nets. We achieve the optimal rate for this class. Third, we give an optimal algorithm for finding stationary points of functions satisfying the Kurdyka-Lojasiewicz (KL) condition. For example, over-parameterized neural networks often satisfy this condition. Fourth, we provide new state-of-the-art rates for stationary points of non-convex population loss functions. Fifth, we obtain improved rates for non-convex generalized linear models. A modification of our algorithm achieves nearly the same rates for second-order stationary points of functions with Lipschitz Hessian, improving over the previous state-of-the-art for each of the above problems.

Probabilistic Methods · Variational Inference

Yunfei Long, Zilin Tian, Liguo Zhang, Huosheng Xu

Mean-field variational inference (MFVI) methods provide computationally cheap approximations to the posterior of Bayesian Neural Networks (BNNs) when compared to alternatives like MCMC. However, applying MFVI to BNNs encounters limitations due to the Monte Carlo sampling problem. This problem stems from two main issues. *First*, most samples do not accurately represent the most probable weights. *Second*, random sampling from variational distributions introduces high variance in gradient estimates, which can hinder the optimization process, leading to slow convergence or even failure. In this paper, we introduce a novel sampling method called *Reparameterized Importance Sampling* (RIS) to estimate the first moment in neural networks, reducing variance during feed-forward propagation. We begin by analyzing the generalized form of the optimal proposal distribution and presenting an inexpensive approximation. Next, we describe the sampling process from the proposal distribution as a transformation that combines exogenous randomness with the variational parameters. Our experimental results demonstrate the effectiveness of the proposed RIS method in three critical aspects: improved convergence, enhanced predictive performance, and successful uncertainty estimation for out-of-distribution data.

Deep Learning · Graph Neural Networks

Songtao Liu, Jinghui Chen, Tianfan Fu, Lu Lin, Marinka Zitnik, Dinghao Wu

This paper introduces a min-max optimization formulation for the Graph Signal Denoising (GSD) problem. In this formulation, we first maximize the second term of GSD by introducing perturbations to the graph structure based on Laplacian distance and then minimize the overall loss of the GSD. By solving the min-max optimization problem, we derive a new variant of the Graph Diffusion Convolution (GDC) architecture, called Graph Adversarial Diffusion Convolution (GADC). GADC differs from GDC by incorporating an additional term that enhances robustness against adversarial attacks on the graph structure and noise in node features. Moreover, GADC improves the performance of GDC on heterophilic graphs. Extensive experiments demonstrate the effectiveness of GADC across various datasets. Code is available at https://github.com/SongtaoLiu0823/GADC.

Deep Learning · Graph Neural Networks

Yang Liu, Deyu Bo, Chuan Shi

The increasing amount of graph data places requirements on the efficient training of graph neural networks (GNNs). The emerging graph distillation (GD) tackles this challenge by distilling a small synthetic graph to replace the real large graph, ensuring GNNs trained on real and synthetic graphs exhibit comparable performance. However, existing methods rely on GNN-related information as supervision, including gradients, representations, and trajectories, which have two limitations. First, GNNs can affect the spectrum (*i.e*., eigenvalues) of the real graph, causing *spectrum bias* in the synthetic graph. Second, the variety of GNN architectures leads to the creation of different synthetic graphs, requiring *traversal* to obtain optimal performance. To tackle these issues, we propose Graph Distillation with Eigenbasis Matching (GDEM), which aligns the eigenbasis and node features of real and synthetic graphs. Meanwhile, it directly replicates the spectrum of the real graph and thus prevents the influence of GNNs. Moreover, we design a discrimination constraint to balance the effectiveness and generalization of GDEM. Theoretically, the synthetic graphs distilled by GDEM are restricted spectral approximations of the real graphs. Extensive experiments demonstrate that GDEM outperforms state-of-the-art GD methods with powerful cross-architecture generalization ability and significant distillation efficiency. Our code is available at https://github.com/liuyang-tian/GDEM.